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Re: Help with Matrix math!by rg0now (Chaplain) 
on Dec 17, 2007 at 19:04 UTC ( #657486=note: print w/ replies, xml )  Need Help?? 
What you are trying to solve is a nonlinearly constrained nonlinear (actually, quadratic) mathematical program. This is a tough one, I can tell you... However, all is not lost, because you might be able to simplify your problem into something actually soluble. First: are you absolutely sure that you want to optimize over the boundary of the unit ball (this is what the constraint x'x=1 seems to impose)? Isn't it enough to only assure that x is constrained into a sane region, like e.g., the ndimensional unit box? If it is, you get a quadratic program: See more on quadratic programs here: Quadratic programming. Second: all depends on whether or not your D is negative definite or not (all eigenvalues are negative). If not, you are out of luck: the problem is (at least) NP hard. If, on the other hand, D is nicely negative definite, then the objective function is concave and the uniqueness of the optimal solution is guaranteed. In this case, you could use a commercial quadratic solver, like CPLEX, or, after suitable linearization, a linear program solver. too. E.g., this one Math::GLPK has a nice Perl interface (believe me it's nice: I wrote it!...:).
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